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We give conditions for when continuous orbit equivalence of one-sided shiftspaces implies flow equivalence of the associated two-sided shift spaces. Usinggroupoid techniques, we prove that this is always the case for shifts of finitetype. This generalises a result of Matsumoto and Matui from the irreducible tothe general case. We also prove that a pair of one-sided shift spaces of finitetype are continuously orbit equivalent if and only if their groupoids areisomorphic, and that the corresponding two-sided shifts are flow equivalent ifand only if the groupoids are stably isomorphic.  As applications we show that two finite directed graphs with no sinks and nosources are move equivalent if and only if the corresponding graph$C^*$-algebras are stably isomorphic by a diagonal-preserving isomorphism (ifand only if the corresponding Leavitt path algebras are stably isomorphic by adiagonal-preserving isomorphism), and that two topological Markov chains areflow equivalent if and only if there is a diagonal-preserving isomorphismbetween the stabilisations of the corresponding Cuntz-Krieger algebras (thelatter generalises a result of Matsumoto and Matui about irreducibletopological Markov chains to a result about general topological Markov chains).  We also show that for general shift spaces, strongly continuous orbitequivalence implies two-sided conjugacy.

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids